# What is the slope of a line perpendicular to the x-axis?

Apr 30, 2018

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#### Explanation:

the slope of a line parallel to the $x$-axis has slope $0$.

the slope of a line perpendicular to another will have a slope which is its negative reciprocal.

the negative reciprocal of a number is $- 1$ divided by the number (e.g. the negative reciprocal of $2$ is $\frac{- 1}{2}$, which is $- \frac{1}{2}$).

the negative reciprocal of $0$ is $- \frac{1}{0}$.

this is undefined, since one cannot define the value of any number that is divided by $0$.

Jun 11, 2018

We say vertical lines have "no slope," horizontal lines have zero slope. The equation is $x = \textrm{c o n s \tan t}$ so it's not equivalent to any slope-intercept form $y = m x + b .$ The slope is undefined because the denominator, change in $x$, is zero.
One may use a direction vector, $\left(p , q\right) ,$ instead of a slope. It's equivalent to a slope $\frac{q}{p}$ but works when $p = 0.$ A line is expressed in parametric form: $\left(x , y\right) = \left(a , b\right) + t \left(p , q\right)$ where $t$ ranges over the reals. The parameter $t$ forms a natural ruler along the line, each increment of one in $t$ is a length $\sqrt{{p}^{2} + {q}^{2}}$ along the line.